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Randomized optimal transport on a graph: framework and new distance measures

Published online by Cambridge University Press:  25 April 2019

Guillaume Guex Open the ORCID record for Guillaume Guex [Opens in a new window] ,
Ilkka Kivimäki  and
Marco Saerens
Guillaume Guex*
Affiliation:
ICTEAM, Universite catholique de Louvain, Louvain-la-Neuve, Belgium (email: [email protected])
Ilkka Kivimäki
Affiliation:
Department of Computer Science, Aalto University, Finland & ICTEAM, Universite catholique de Louvain, Belgium (email: [email protected])
Marco Saerens
Affiliation:
ICTEAM, Universite catholique de Louvain & Universite Libre de Bruxelles, Belgium (email: [email protected])
*
*Corresponding author. Email: [email protected]
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Abstract

The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs–Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature T) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins).When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths and require the inversion of an n × n matrix with n being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T → 0+) and the solution to the corresponding electrical circuit (T → ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.

Keywords

network scienceoptimal transportationbag-of-pathsrandomized shortest pathdistances between nodeslink analysis
Type
Original Article
Information
Network Science , Volume 7 , Issue 1 , March 2019 , pp. 88 - 122
Copyright
© Cambridge University Press 2019 

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